Approaches to expanding the two-arm biased coin randomization to unequal allocation while preserving the unconditional allocation ratio.

The paper discusses three methods for expanding the biased coin randomization (BCR) to unequal allocation while preserving the unconditional allocation ratio at every step. The first method originally proposed in the contexts of BCR and minimization is based on mapping from an equal allocation multi-arm BCR. Despite the improvement proposed in this paper to ensure tighter adherence to the targeted unequal allocation, this method still distributes the probability mass at least as wide as the permuted block randomization (PBR). This works for smaller block sizes, but for larger block sizes, a tighter control of the imbalance in the treatment assignments is desired. The second method, which has two versions, allows to tighten the distribution of the imbalance compared with that achieved with the PBR. However, the distribution of the imbalance remains considerably wider than that of the brick tunnel randomization - the unequal allocation procedure with the tightest possible imbalance distribution among all allocation ratio preserving procedures with the same allocation ratio. Finally, the third method, the BCR with a preset proportion of maximal forcing, mimics the properties of the equal allocation BCR. With maximum forcing, it approaches the brick tunnel randomization, similar to how 1:1 BCR approaches 1:1 PBR with the permuted block size of 2 (the equal allocation procedure with the lowest possible imbalance) when the bias approaches 1. With minimum forcing, the BCR with a preset proportion of maximal forcing approaches complete randomization (similar to 1:1 BCR). Copyright © 2017 John Wiley & Sons, Ltd.